Lecture Notes in Mathematics 2061 Editors: J.-M. Morel, Cachan B. Teissier, Paris For further volumes: http://www.springer.com/series/304 Levy´ in Stanford (Permission granted by G.L. Alexanderson) “Levy´ Matters” is a subseries of the Springer Lecture Notes in Mathematics, devoted to the dissemination of important developments in the area of Stochastics that are rooted in the theory of Levy´ processes. Each volume will contain state-of-the-art theoretical results as well as applications of this rapidly evolving field, with special emphasis on the case of discontinuous paths. Contributions to this series by leading experts will present or survey new and exciting areas of recent theoretical developments, or will focus on some of the more promising applications in related fields. In this way each volume will constitute a reference text that will serve PhD students, postdoctoral researchers and seasoned researchers alike. Editors Ole E. Barndorff-Nielsen Jean Jacod Thiele Centre for Applied Mathematics Institut de Mathematiques´ de Jussieu in Natural Science CNRS-UMR 7586 Department of Mathematical Sciences Universite´ Paris 6 - Pierre et Marie Curie Aarhus University 75252 Paris Cedex 05, France 8000 Aarhus C, Denmark [email protected] [email protected] Jean Bertoin Claudia Kl¨uppelberg Institut f¨ur Mathematik Zentrum Mathematik Universitat¨ Z¨urich Technische Universitat¨ M¨unchen 8057 Z¨urich, Switzerland 85747 Garching bei M¨unchen, Germany [email protected] [email protected] Managing Editors Vicky Fasen Robert Stelzer Department of Mathematics Institute of Mathematical Finance ETH Z¨urich Ulm University 8092 Z¨urich, Switzerland 89081 Ulm, Germany [email protected] [email protected] The volumes in this subseries are published under the auspices of the Bernoulli Society. Serge Cohen • Alexey Kuznetsov Andreas E. Kyprianou • Victor Rivero Levy´ Matters II Recent Progress in Theory and Applications: Fractional Levy´ Fields, and Scale Functions 123 Serge Cohen Andreas E. Kyprianou Universite´ Paul Sabatier University of Bath Institut Mathematique´ de Toulouse Department of Mathematical Sciences Toulouse Cedex 4, France Bath, United Kingdom Alexey Kuznetsov Victor Rivero York University CIMAT A.C. Department of Mathematics Guanajuato, Col. Valenciana, Mexico and Statistics Toronto, ON, Canada ISBN 978-3-642-31406-3 ISBN 978-3-642-31407-0 (eBook) DOI 10.1007/978-3-642-31407-0 Springer Heidelberg New York Dordrecht London Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2012945523 Mathematics Subject Classification (2010): Primary: 60G10, 60G40, 60G51, 60G70, 60J10, 60J45 Secondary: 91B28, 91B70, 91B84 c Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. 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Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface This second volume of the series “Levy´ Matters” consists of two surveys of two topical areas, namely Fractional Levy´ Fields by Serge Cohen and the Theory of Scale Functions for Spectrally Negative Levy´ Processes by Alexey Kuznetsov, Andreas Kyprianou and Victor Rivero. Roughly speaking, irregularity is a crucial aspect of random phenomena that appears in many different contexts. An important issue in this direction is to offer tractable mathematical models that encompass the variety of observed behaviours in applications. Fractional Levy´ fields are constructed by integration of Levy´ random measures; somehow they interpolate between Gaussian and stable random fields. They exhibit a number of interesting features including local asymptotic self- similarity and multi-fractional aspects. Calibration techniques and simulation of fractional Levy´ fields constitute important elements for many applications. A real-valued Levy´ process is spectrally negative when it has no positive jumps. In this situation, the distribution of several variables related to the first exit-time from a bounded interval can be specified in terms of the so-called scale functions; the latter also play a fundamental role in other aspects of the theory. Scale functions are characterized by their Laplace transform, but in general no explicit formula is known, and therefore it is crucial in many applications to gather information about their asymptotic behaviour and regularity and to provide efficient numerical methods to compute them. Aarhus, Denmark Ole E. Barndorff-Nielsen Z¨urich, Switzerland Jean Bertoin Paris, France Jean Jacod Munich, Germany Claudia K¨uppelberg v • From the Preface to Levy´ Matters I Over the past 10–15 years, we have seen a revival of general Levy´ processes theory as well as a burst of new applications. In the past, Brownian motion or the Poisson process had been considered as appropriate models for most applications. Nowa- days, the need for more realistic modelling of irregular behaviour of phenomena in nature and society like jumps, bursts and extremes has led to a renaissance of the theory of general Levy´ processes. Theoretical and applied researchers in fields as diverse as quantum theory, statistical physics, meteorology, seismology, statistics, insurance, finance and telecommunication have realized the enormous flexibility of Levy´ models in modelling jumps, tails, dependence and sample path behaviour. Levy´ processes or Levy-driven´ processes feature slow or rapid structural breaks, extremal behaviour, clustering and clumping of points. Tools and techniques from related but distinct mathematical fields, such as point processes, stochastic integration, probability theory in abstract spaces and differential geometry, have contributed to a better understanding of Levy´ jump processes. As in many other fields, the enormous power of modern computers has also changed the view of Levy´ processes. Simulation methods for paths of Levy´ processes and realizations of their functionals have been developed. Monte Carlo simulation makes it possible to determine the distribution of functionals of sample paths of Levy´ processes to a high level of accuracy. This development of Levy´ processes was accompanied and triggered by a series of Conferences on Levy´ Processes: Theory and Applications. The First and Second Conferences were held in Aarhus (1999, 2002), the Third in Paris (2003), the Fourth in Manchester (2005) and the Fifth in Copenhagen (2007). To show the broad spectrum of these conferences, the following topics are taken from the announcement of the Copenhagen conference: • Structural results for Levy´ processes: distribution and path properties •Levy´ trees, superprocesses and branching theory • Fractal processes and fractal phenomena • Stable and infinitely divisible processes and distributions vii viii From the Preface to Levy´ Matters I • Applications in finance, physics, biosciences and telecommunications •Levy´ processes on abstract structures • Statistical, numerical and simulation aspects of Levy´ processes •Levy´ and stable random fields At the Conference on Levy´ Processes: Theory and Applications in Copenhagen the idea was born to start a series of Lecture Notes on Levy´ processes to bear witness of the exciting recent advances in the area of Levy´ processes and their applications. Its goal is the dissemination of important developments in theory and applications. Each volume will describe state-of-the-art results of this rapidly evolving subject with special emphasis on the non-Brownian world. Leading experts will present new exciting fields, or surveys of recent developments, or focus on some of the most promising applications. Despite its special character, each article is written in an expository style, normally with an extensive bibliography at the end. In this way each article makes an invaluable comprehensive reference text. The intended audience are PhD and postdoctoral students, or researchers,